For a Köthe sequence space, the classes of Λ0-nuclear spaces and spaces with the Λ0-property are introduced and studied and the relation between them is investigated. Also, we show that, for Λ0 6= c0, these classes of spaces are in general different from the corresponding ones for Λ0 = c0, which have been extensively studied in the non-archimedean literature (see, for example, [1]–[6]).